In communications networks there are source signals intended for a specific communications device, and there are source signals intended for other communications devices operating within the same frequency band. There are also sources of noise which produce signals that are not used for communications, but are received by the communications devices as well.
To facilitate decoding of the source signals of interest, signal separation is used to separate the signals received by a communications device. If the signals can be separated without any knowledge about the nature of the signals or the transformations that occur due to interactions between the signals and the communication channel, then this is known as blind signal separation In practical implementations, any knowledge that is available is often exploited In this case, the signal separation is semi-blind or non-blind. When the processing is non-blind, signal extraction techniques are often used instead of separation The difference being that more of the interferers are treated as noise than in the separation procedures
In blind signal separation techniques, the received signals are often compactly represented by matrix equations of the form:x=As+n  Equation 1where x is the received signal vector, A is the mixing matrix, s is the received vector composed of desired and undesired signals which are separable by signal processing, and n is an aggregate noise vector composed of random noise sources and any actual signal not included in the s vector
The mixing matrix A and signals remain combined in the received signal vector s such thaty=W(As)+Wn=Wx  Equation 2where W is the separation matrix, and y is a vector that is a subset of s in an unknown order with scaling changes. If all the signals are not separable, then the noise term n includes the residual signal due to the unidentifiable sources.
In semi-blind or non-blind signal separation techniques, A is referred to as the channel matrix, and the signal vector s may be solved by determining the inverse channel matrix:s′=A−1x=A−1(As)+A−1n=s+A−1n  Equation 3s′ is therefore the signal vector of interest plus noise multiplied by the inverse channel matrix. Traditional techniques may be used to find the inverse of the channel.
For purposes of discussion, the channel matrix and the mixing matrix will be generally referred to herein simply as the matrix. Regardless of whether the matrix is a mixing matrix for blind signal separation or a channel matrix for non-blind/semi-blind signal separation, the rank of the matrix determines how many signals can actually be separated. For example, a matrix having a rank of 4 means that 4 source signals can be separated. Ideally, the rank of the matrix should at least be equal to the number of signal sources. The larger the rank, the more signals that can be separated.
If there is an equal number of linear equations in the matrix as there are unknowns, then the matrix is said to be rank sufficient. To solve a rank sufficient matrix, an L0-norm technique can be used to provide the unique solution. This is the ideal case in which the matrix is adequately populated so that the number of unknown variables is equal to the number of linear equations.
However, when the matrix contains more unknown variables than linear equations, the matrix is underdetermined. In an underdetermined matrix, various combinations of the variables may be utilized to satisfy the matrix constraints. In this situation, there are infinitely many solutions,
One approach to solving a rank deficient matrix is to increase the rank of the matrix. U.S. published patent application No. 2006/0066481 discloses several such techniques for increasing the rank of the matrix. This patent is assigned to the current assignee of the present invention, and is incorporated herein by reference in its entirety. To increase the rank of the matrix associated with a communications device, more antennas may be added to populate the matrix with additional linearly independent signal sums. However, small portable communications devices have little available volume for a large number of antennas, and mounting the antennas on the outside of the communications devices is a problem for the users.
Consequently, there is a need to separate signals from an underdetermined matrix without increasing the number of antennas. Research by David Donoho of Stanford University concludes that for most underdetermined systems of equations, when a sufficiently sparse solution exits, it can be found by convex optimization. More precisely, for a given ratio of unknowns (m) to equations (n), there is a threshold p so that most large n by m matrices generate systems of equations with two properties: 1) if convex optimization is run to find an L1-norm minimal solution, and the solution has fewer than pn non-zeros, then this is the unique sparsest solution to the equations, and 2) if the result does not have pn non-zeros, there is no solution with <pn non-zeros.
Donoho has shown that under specific conditions rank deficient matrices can be uniquely solved. However, a problem remains that while the number of solutions may be reduced to a finite number or at least a more constrained set, a unique solution may still not be obtained.